Lesson #1: Trigonometric Ratios: A Student Investigation
Grade/Content Area
Physics 1 (H)
Lesson Title
Trigonometric Ratios: A Student Investigation
State Standards: GLEs/GSEs National Content Standards:
Grade Span Expectations:
M(N&O)–10–8 Applies properties of numbers to solve problems, to simplify computations, or to compare and contrast the properties of numbers and number systems.
M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts.
M(F&A)–10–1 Identifies, extends, and generalizes a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs to solve problems.
M(PRP)–HS–1 Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content and be able to:
• Expand the repertoire of problem-solving strategies and use those strategies in more sophisticated ways.
• Use technology whenever appropriate to solve real-world problems (e.g., personal finance, wages, banking and credit, home improvement problems, measurement, taxes, business situations, purchasing, and transportation).
• Formulate and redefine problem situations as needed to arrive at appropriate conclusions.
Common Core Standards:
Define trigonometric ratios and solve problems involving right triangles
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
Context of the Lesson Where does this lesson fit in the curriculum and instructional context? Is it the opening of a unit or a series of lessons?
This lesson will take place in a Physics 1 or Physics 1 (H) classroom. The learning abilities of the students in my class are fairly diverse. The main difference can be seen in their conceptual understanding of basic mathematical principles. Some students have a very strong mathematical foundation while others are not so confident with the algebraic knowledge that is required for this course.
This lesson is designed to be the first unit of the Projectile Motion unit. This lesson may be the first exposure that the students will have had to sine, cosine, and tangent. If so than the first portion of the lesson will be performed along with the second section. If students have already learned about the basic trigonometric ratios then the first section will be skipped. We will focus on trigonometric ratios for acute angles in a right triangle. Many times students taking physics have not yet been exposed to sine, cosine, and tangent and thus this lesson would be beneficial to cover before studying vectors and/or two-dimensional kinematics. In order to find the components of a given vector in 2-D space, students need to be able to use the basic trigonometric ratios for acute angles in a right triangle.
Prior to this lesson, students should already have been exposed to the idea of a function. They should also know how to find the value of a function for a given value of the domain. Students should also be able to manipulate equations that include 1 unknown and they should be familiar with working with ratios. In addition, students should have already been exposed to basic facts about right triangles including the Pythagorean theorem and basic terminology. With this prerequisite knowledge, students should be successful during this lesson.
Opportunities to Learn
Differentiation: Materials, Learners and Environments
Plans to differentiate instruction: This lesson includes several different forms of instruction so as to maximize the learning experience for all of my students. The lesson includes direct instruction segments, group work, hands-on learning activities, and inquiry based discovery learning. Therefore the lesson will be beneficial for the visual and kinesthetic learners as well as the students that learn best through direct instruction. In addition to this, throughout the lesson (as I mentioned above) I will have “extra” challenges for the advanced students. During the first activity, if students finish early I will have them consider the sec/csc/cot buttons on their calculators as well. During the second activity, if students finish early I will have other challenges ready for them that will be more challenging than the first. In this way I will allow the advanced students to learn at their own pace and not be held back by the pace of the majority. For the students that may be struggling I will have different levels of supports that I offer them throughout the activities. These supports are explained in more detail in the next section. With inquiry based activities it is important to not provide too much support so I have carefully thought about how much information I can give the students without turning the activity into a direct instruction activity. I will also explain to my students that if they would like to stay after school to finish any of the activities I would be able to stay after and perform them with them to ensure that they fully understand the underlying concepts. Accommodations and modifications: All classes (at all levels) will consist of a heterogeneous group of students. Every student in the class will learn at a different pace and will have different strengths and weaknesses and their “zones of proximal development” will differ from one another. Each student will also have different learning styles. It is important to make modifications and accommodations to any lesson to accommodate for the advanced students in the class as well as for the students that may be struggling. In order to do this during this lesson I will be sure to choose groups that consist of students that have similar “zones of proximal development”. By this I mean that I will place students together that learn at similar paces. In this way, each of the groups can work at their own pace and no one in the group will feel as though they are moving too slowly or too quickly or that they are completing the majority of the work. By grouping students in this was I would also be able to offer extra help to the struggling groups and provide extra challenges to the groups that are finishing the tasks easily. I will also try to group students (of similar learning ability) together that have different learning styles. In this way the visual learners can learn from the mathematical and kinesthetic learners and vice-versa. Environment factors: This lesson involves two separate activities. The first activity will be held in the classroom and students will be working in groups. To make this group work easier for the students I will have already organized the desks into groups. By doing this I provide each group with a designated area where they can work and I preserve the “flow” of the lesson since students don’t have to shuffle their desks during the lesson. The materials for the activity will be located at the front of the room and will be easily accessible to all of the groups so that there isn’t a backup of “traffic” in the classroom. The second activity will be held on the football field. This will require the permission of administration and possibly the permission of parents. I will stand in a position of the field so that I can see all of the students and they can all see me. This will keep them all on task and focused. In the event of weather issues I will have to alter this activity and either perform it in the gymnasium or postpone it. If the physical education classes are out on the field, the lesson may become a little more difficult. The activity could be moved to a vacant soccer field if necessary. Materials: ü Protractors ü Measuring Tapes/ Rulers ü Calculators ü Foam Board (3 pieces)
Objectives
Students will demonstrate their understanding of basic trigonometric ratios by successfully determining how the sine, cosine, and tangent of a given acute angle of a right triangle are related to the sides of the triangle.
Students will demonstrate their understanding of the relationships that exist between the angles and sides of a right triangle by successfully splitting “vectors” into their horizontal and vertical components and following navigation directions on the football field.
Instructional Procedures
This particular lesson will include two “LES” procedures, which will be detailed below. The first LES procedure corresponds to objective (i) while the second corresponds to objectives (ii) seen in the above section. LAUNCH A: The first part of this lesson is designed to introduce students to sine, cosine, and tangent. Before delving into the subject matter for the day I will provide the students with a short problem that involves using the Pythagorean theorem. “Today I have a few challenges for all of you and we are going to learn more about right triangles. But before we get started with the investigation I would like everyone to complete the “daily challenge”. You can work in pairs if you’d like” This will be our “Daily Challenge” for the day. I will perform this activity to ensure that students are aware of the Pythagorean theorem and have at least briefly studied right triangles in the past. The problem that I will use will involve the Pythagorean theorem but it will not be an obvious application of it. This document is attached and is titled Lesson 1 [A]. This activity should help the students begin to realize that math is all around us and can be used to solve everyday problems. Students must also understand what is meant by “hypotenuse” and what is meant by “opposite” and “adjacent” sides to an angle in a given right triangle. I will use this first activity to review these terms with them and test their prior knowledge. I will also make sure that they all understand that a triangle can be reoriented and spun in any direction and ensure that they are still able to identify the various sides of the triangle. If students are unable to see that they need to use the Pythagorean theorem in this situation I will offer some guidance: “Think about what you have learned in the past about the relationships between the sides and the hypotenuse of a right triangle. A very important mathematician developed a rule that is very useful in these types of situations.” Once we have reviewed the Pythagorean theorem and the “Daily Challenge” I will instruct all of the students to take out a calculator.“Okay class, I would like everyone to take out your calculators! Has anyone ever wondered what the “sin”, “cos”, and “tan” buttons on the calculator actually do? We are going to find out today!” If they do not have calculators I will have calculators that they can borrow from me for the class period. I will then inform the students that I would like them to find the “sin”, “cos” and “tan” buttons on the calculator and I will ask them if they are familiar with the use of those buttons. Of course they won’t be because they have not yet been exposed to them. I will briefly explain to them that “sin” “cos” and “tan” represent the functions sine, cosine, and tangent. I will also explain to them that the sine, cosine, and tangent of an acute angle in a right triangle are related to the lengths of the sides of the triangle. I will not explain that they represent ratios of certain sides of the triangle but rather I will just explain that they are “related” in some way to the sides of the triangle. “We are now going to work in groups on the second challenge of the day! The groups can be seen on the whiteboard and the groups stations have already been labeled.” I will then pass out the second “Challenge” to my students. This document is attached and is titled Lesson 1 [B]. For a right triangle with an angle ß, the challenge is for students to determine how sin(ß), cos(ß) and tan(ß) are related to the length of the sides of the triangle. I will construct different right triangles from foam board and students will have to measure the sides and the angles of the triangle and experiment to determine the relationship between the sine, cosine, and tangent of an angle and the lengths of the sides. On the “Challenge” sheet I provide them with a few clues to help them in this process. Other than the clues that I provide them, the assignment is very open-ended and students can go in a number of directions. For this challenge I will have previously assigned groups (as was discussed in the above section). NOTE: All lengths of the sides of the triangles will be whole numbers to allow for easier identification of the relationships that exist. I will construct several triangles that are “similar”. By this I mean that the three interior angles of several of the triangles will be the same and the only difference will be the size of the triangle. This will help students to understand that sine, cosine, and tangent are only dependent on the given angle and represent ratios of the sides. If two triangles are similar, the fact that one of the triangles is larger does not affect the ratio and thus does not affect the value of sine, cosine, or tangent.
EXPLORE A: After introducing the activity to the students I will allow them to break up into groups and begin to explore with the various triangles that I will provide to them. If they feel as though they found a correct relationship I will make sure that they test out their hypotheses on the other triangles as well. I will circulate the classroom and ensure that all groups are on the right track and aren’t overly frustrated with the task. I will keep them in the “meadow”. If groups are having trouble with the task I will offer some guidance: “How can we approach this task in a more organized way rather than simply guessing and checking?” “What can you tell me about the “sine” of one of the angles of this triangle? Is it larger or smaller than the legs of the triangle?” “You may want to try dividing the lengths of two of the sides of the triangle?” If groups believe that they have found the answer I will question them further. “Have you checked your results for several (3 or 4) triangles?” “Why does the size of the triangle not affect the sine, cosine, or tangent of the angle?” “If I knew the measure of an angle in a right triangle and the length of the hypotenuse, can I determine anything else about the sides of the triangle?” “What can you tell me about the csc, sec, and cot buttons on the calculator?” If certain groups are really struggling I will have two different forms of support that I will offer to them. The first is simply a table that will help them to organize the values and allow them to see the relationships between the numbers more clearly. This table can be seen in the attached document titled Lesson 1 [C]. Ideally I would like for them to develop this table on their own but I will have one prepared just in case. This will help the students organize their thoughts and figure out exactly what they are comparing. If groups are still struggling I will provide them with the second level of support. I will give them a page that shows the possible relationships that they can find between the sides of the triangle and the sin/cos/tan of a given angle. This can be seen in the attached document Lesson 1 [D]. SUMMARIZE/SHARE A: Once all of the groups have finished “exploring” the trigonometric relationships, the class will reconvene as a whole. “Okay class! It seems as though everyone has completed the challenge. However, it’s now time to figure out which relationships are the correct relationships! At each group’s station there should be a small whiteboard. I want you to record your findings on this whiteboard and choose a spokesperson from your group who will have to present to the class and convince us that your answers are correct!” I will have each group present their findings and attempt to convince the rest of the class that the relationships that they found are the correct relationships. Each group will have been given a small whiteboard on which they can record their findings and they can have a spokesperson use this whiteboard to explain to the class what they found out and the process that they used. This will lead to a class discussion in which each group displays their findings and compares them to the rest of the class. As a whole, we will come to a conclusion about the proper relationships and I will explain how this leads to the SOHCAHTOA acronym. Ideally students will be able to establish the three basic ratios and I will present this acronym to them in order to give them an easier way to remember the ratios. I will formalize these rules and present the students with a brief note session. After this activity they should fully understand how the sides of a right triangle are related to the cos/sine/tan of the acute angles of the triangle. LAUNCH B: The second LES segment that will occur throughout this lesson includes a “scavenger” hunt on the football field. I will explain to the students that we will be going outside for a scavenger hunt and that they will need their notebooks, a tape measure, and a calculator. Prior to going outside I will explain what is meant by “45 degrees North of East”. Many directions are expressed in this way and I want to ensure that they understand the idea. Next, I will have the students find a partner and each group of partners will receive an envelope. “Okay class! Now we will be taking a little trip out to the football field! Each group of partners will receive an envelope with a set of directions that you must follow on the field. You will have to apply what you just learned in the previous activity in order to follow the directions” On the front of the envelope will be a location on the football field from which they must begin. This location will differ for all of the groups. They will not be able to open the envelope until they get to the football field. The students should all be intrigued by this activity and curious to see what’s in the envelope. EXPLORE B: Inside each of the student’s envelopes will be a set of 10 directions for them to follow. The students must follow these 10 directions starting from the designated location and once they are done they must record what yard line they are closest to (specifying which side of the field as well) and the distance they are from the closest sideline. They will bring this information back to me and I will check to see if they finished in the correct location (i.e. if they followed the directions properly) I will have calculated where each envelope should end up in order to make this process quicker. If they did not finish in the correct location, then they must repeat the sequence to determine where they made a mistake. The directions will be similar to the following (walk 10 yards due north OR walk 15 yards @ 30 degrees North of East). Since the students do not have a protractor they can’t walk in a diagonal line. Therefore, they must find the components of the given “vectors” in the direction parallel with the yard lines and perpendicular to the yard lines (x and y). To do this they need to use what they learned about sine and cosine in the first portion of the lesson. An example of a set of directions can be seen in the document Lesson 1 [E]. If students are struggling I will ask them some guiding questions: “If you do not have a protractor to measure out an angle how can we accurately determine where to move?” “Remember that we can walk parallel and perpendicular to the field without needing to use a protractor” “With the angle and the distance that is given in each of the directions is there any way to form a right triangle?” “Which side of the right triangle does the distance that is given represent?” NOTE: For simplicity, “north” will be specified to be in the direction of the field goal that is closest to the true north direction. I will make this clear to my students. If groups finish early I will have more challenging envelopes for them to try and they can also exchange envelopes to get more practice. If groups are really struggling with this assignment I will follow the directions with them to ensure that they are using the proper equations and techniques. I may also pair these groups up with groups that were successful and have them assist one another. SUMMARIZE/SHARE B: Each group will return to me on the field once they finish an envelope and I will provide them with a new one. Once I feel as though all of the students completed the activity and found it beneficial we will return to the classroom and talk about the process that was used throughout the investigation. I will ask them what they found challenging and I will clear up any misconceptions that students may have had.
Assessment
In order to assess my students during this investigation I will determine whether each student has successfully completed each of the objectives described above. Launch A: During the Launch phase of the first activity, the student will be informally assessed and I will determine whether or not they are prepared to begin working on the first investigation. I will walk around to the different groups during this first activity and closely observe the work that they are doing. If they seem confused on dealing with right triangles and on identifying the various sides of a right triangle (especially when the triangle is positioned differently) then I may need to spend more time in this launch phase in order to build up the prerequisite knowledge that will be necessary for the lesson. Explore A:
During the first investigation I will again assess the students informally on their understanding of the content of the lesson. I will observe each of the groups during this investigation and I will ask them the questions that I have detailed above. Based on their responses and on their ability to complete the student investigation, I will be able to determine how well they understand the content of the lesson and how prepared they will be to apply these concepts to other applications. I will keep a journal during the class period and record notes about each student in the class to ensure that they understand the concepts that are being investigated and to ensure that they have the necessary skills that will be required to move forward in the curriculum. I want to ensure that all of the students demonstrate an understanding about how the sine, cosine, and tangent of an acute angle in a right triangle are related to the sides of the triangle. No grade will be assigned to each student but I will use the assessments that I make to determine whether or not more instruction of the concepts will be necessary. If the class as a whole struggled with the material I will plan a follow up lesson to help them understand the material more fully. Summarize A:
During the summarize phase of the first activity the students will have to explain their thought process as well as their results. This will give me an opportunity to further gauge their understanding of the material. If a student can fully explain what they found and how it relates to right triangles in general then I can conclude that they have a firm grasp on the necessary concepts. I can also assess the students in a formal way by determining if the results that they found are accurate.Launch/Explore/Summarize B:
During this second activity I will informally assess the students on their ability to apply what they learned in the first activity to a real world application. I will observe them as they follow the directions given to them on the football field and I will determine if they truly understand how to use sine, cosine, and tangent to determine information about a given right triangle (break a vector into its components). Again, if I do not feel as though the class (as a whole) is proficient at the task then I will certainly prepare a follow up exercise to provide the students with more practice on the material. I can also assess them formally by determining if they completed the set of directions properly and wound up in the location that they were supposed to finish in.
Reflections** This section to be completed only if lesson plan is implemented.
Student Work Sample 1 – Approaching Proficiency:
Student Work Sample 2 – Proficient:
Student Work Sample 3 – Exceeds Proficiency:
Lesson Implementation: Was not able to implement this lesson yet.
Lesson #1: Trigonometric Ratios: A Student Investigation
Lesson Title
Trigonometric Ratios: A Student Investigation
National Content Standards:
M(N&O)–10–8 Applies properties of numbers to solve problems, to simplify computations, or to compare and contrast the properties of numbers and number systems.
M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts.
M(F&A)–10–1 Identifies, extends, and generalizes a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs to solve problems.
M(PRP)–HS–1 Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content and be able to:
• Expand the repertoire of problem-solving strategies and use those strategies in more sophisticated ways.
• Use technology whenever appropriate to solve real-world problems (e.g., personal finance, wages, banking and credit, home improvement problems, measurement, taxes, business situations, purchasing, and transportation).
• Formulate and redefine problem situations as needed to arrive at appropriate conclusions.
Common Core Standards:
Define trigonometric ratios and solve problems involving right triangles
Where does this lesson fit in the curriculum and instructional context? Is it the opening of a unit or a series of lessons?
This lesson is designed to be the first unit of the Projectile Motion unit. This lesson may be the first exposure that the students will have had to sine, cosine, and tangent. If so than the first portion of the lesson will be performed along with the second section. If students have already learned about the basic trigonometric ratios then the first section will be skipped. We will focus on trigonometric ratios for acute angles in a right triangle. Many times students taking physics have not yet been exposed to sine, cosine, and tangent and thus this lesson would be beneficial to cover before studying vectors and/or two-dimensional kinematics. In order to find the components of a given vector in 2-D space, students need to be able to use the basic trigonometric ratios for acute angles in a right triangle.
Prior to this lesson, students should already have been exposed to the idea of a function. They should also know how to find the value of a function for a given value of the domain. Students should also be able to manipulate equations that include 1 unknown and they should be familiar with working with ratios. In addition, students should have already been exposed to basic facts about right triangles including the Pythagorean theorem and basic terminology. With this prerequisite knowledge, students should be successful during this lesson.
Differentiation: Materials, Learners and Environments
This lesson includes several different forms of instruction so as to maximize the learning experience for all of my students. The lesson includes direct instruction segments, group work, hands-on learning activities, and inquiry based discovery learning. Therefore the lesson will be beneficial for the visual and kinesthetic learners as well as the students that learn best through direct instruction.
In addition to this, throughout the lesson (as I mentioned above) I will have “extra” challenges for the advanced students. During the first activity, if students finish early I will have them consider the sec/csc/cot buttons on their calculators as well. During the second activity, if students finish early I will have other challenges ready for them that will be more challenging than the first. In this way I will allow the advanced students to learn at their own pace and not be held back by the pace of the majority.
For the students that may be struggling I will have different levels of supports that I offer them throughout the activities. These supports are explained in more detail in the next section. With inquiry based activities it is important to not provide too much support so I have carefully thought about how much information I can give the students without turning the activity into a direct instruction activity. I will also explain to my students that if they would like to stay after school to finish any of the activities I would be able to stay after and perform them with them to ensure that they fully understand the underlying concepts.
Accommodations and modifications:
All classes (at all levels) will consist of a heterogeneous group of students. Every student in the class will learn at a different pace and will have different strengths and weaknesses and their “zones of proximal development” will differ from one another. Each student will also have different learning styles. It is important to make modifications and accommodations to any lesson to accommodate for the advanced students in the class as well as for the students that may be struggling. In order to do this during this lesson I will be sure to choose groups that consist of students that have similar “zones of proximal development”. By this I mean that I will place students together that learn at similar paces. In this way, each of the groups can work at their own pace and no one in the group will feel as though they are moving too slowly or too quickly or that they are completing the majority of the work. By grouping students in this was I would also be able to offer extra help to the struggling groups and provide extra challenges to the groups that are finishing the tasks easily. I will also try to group students (of similar learning ability) together that have different learning styles. In this way the visual learners can learn from the mathematical and kinesthetic learners and vice-versa.
Environment factors:
This lesson involves two separate activities. The first activity will be held in the classroom and students will be working in groups. To make this group work easier for the students I will have already organized the desks into groups. By doing this I provide each group with a designated area where they can work and I preserve the “flow” of the lesson since students don’t have to shuffle their desks during the lesson. The materials for the activity will be located at the front of the room and will be easily accessible to all of the groups so that there isn’t a backup of “traffic” in the classroom.
The second activity will be held on the football field. This will require the permission of administration and possibly the permission of parents. I will stand in a position of the field so that I can see all of the students and they can all see me. This will keep them all on task and focused. In the event of weather issues I will have to alter this activity and either perform it in the gymnasium or postpone it. If the physical education classes are out on the field, the lesson may become a little more difficult. The activity could be moved to a vacant soccer field if necessary.
Materials:
ü Protractors
ü Measuring Tapes/ Rulers
ü Calculators
ü Foam Board (3 pieces)
Instructional Procedures
LAUNCH A:
The first part of this lesson is designed to introduce students to sine, cosine, and tangent. Before delving into the subject matter for the day I will provide the students with a short problem that involves using the Pythagorean theorem.
“Today I have a few challenges for all of you and we are going to learn more about right triangles. But before we get started with the investigation I would like everyone to complete the “daily challenge”. You can work in pairs if you’d like”
This will be our “Daily Challenge” for the day. I will perform this activity to ensure that students are aware of the Pythagorean theorem and have at least briefly studied right triangles in the past. The problem that I will use will involve the Pythagorean theorem but it will not be an obvious application of it. This document is attached and is titled Lesson 1 [A]. This activity should help the students begin to realize that math is all around us and can be used to solve everyday problems. Students must also understand what is meant by “hypotenuse” and what is meant by “opposite” and “adjacent” sides to an angle in a given right triangle. I will use this first activity to review these terms with them and test their prior knowledge. I will also make sure that they all understand that a triangle can be reoriented and spun in any direction and ensure that they are still able to identify the various sides of the triangle.
If students are unable to see that they need to use the Pythagorean theorem in this situation I will offer some guidance:
“Think about what you have learned in the past about the relationships between the sides and the hypotenuse of a right triangle. A very important mathematician developed a rule that is very useful in these types of situations.”
Once we have reviewed the Pythagorean theorem and the “Daily Challenge” I will instruct all of the students to take out a calculator.“Okay class, I would like everyone to take out your calculators! Has anyone ever wondered what the “sin”, “cos”, and “tan” buttons on the calculator actually do? We are going to find out today!”
If they do not have calculators I will have calculators that they can borrow from me for the class period. I will then inform the students that I would like them to find the “sin”, “cos” and “tan” buttons on the calculator and I will ask them if they are familiar with the use of those buttons. Of course they won’t be because they have not yet been exposed to them. I will briefly explain to them that “sin” “cos” and “tan” represent the functions sine, cosine, and tangent. I will also explain to them that the sine, cosine, and tangent of an acute angle in a right triangle are related to the lengths of the sides of the triangle. I will not explain that they represent ratios of certain sides of the triangle but rather I will just explain that they are “related” in some way to the sides of the triangle.
“We are now going to work in groups on the second challenge of the day! The groups can be seen on the whiteboard and the groups stations have already been labeled.”
I will then pass out the second “Challenge” to my students. This document is attached and is titled Lesson 1 [B]. For a right triangle with an angle ß, the challenge is for students to determine how sin(ß), cos(ß) and tan(ß) are related to the length of the sides of the triangle. I will construct different right triangles from foam board and students will have to measure the sides and the angles of the triangle and experiment to determine the relationship between the sine, cosine, and tangent of an angle and the lengths of the sides. On the “Challenge” sheet I provide them with a few clues to help them in this process. Other than the clues that I provide them, the assignment is very open-ended and students can go in a number of directions.
For this challenge I will have previously assigned groups (as was discussed in the above section).
NOTE: All lengths of the sides of the triangles will be whole numbers to allow for easier identification of the relationships that exist. I will construct several triangles that are “similar”. By this I mean that the three interior angles of several of the triangles will be the same and the only difference will be the size of the triangle. This will help students to understand that sine, cosine, and tangent are only dependent on the given angle and represent ratios of the sides. If two triangles are similar, the fact that one of the triangles is larger does not affect the ratio and thus does not affect the value of sine, cosine, or tangent.
EXPLORE A:
After introducing the activity to the students I will allow them to break up into groups and begin to explore with the various triangles that I will provide to them. If they feel as though they found a correct relationship I will make sure that they test out their hypotheses on the other triangles as well. I will circulate the classroom and ensure that all groups are on the right track and aren’t overly frustrated with the task. I will keep them in the “meadow”.
If groups are having trouble with the task I will offer some guidance:
“How can we approach this task in a more organized way rather than simply guessing and checking?”
“What can you tell me about the “sine” of one of the angles of this triangle? Is it larger or smaller than the legs of the triangle?”
“You may want to try dividing the lengths of two of the sides of the triangle?”
If groups believe that they have found the answer I will question them further.
“Have you checked your results for several (3 or 4) triangles?”
“Why does the size of the triangle not affect the sine, cosine, or tangent of the angle?”
“If I knew the measure of an angle in a right triangle and the length of the hypotenuse, can I determine anything else about the sides of the triangle?”
“What can you tell me about the csc, sec, and cot buttons on the calculator?”
If certain groups are really struggling I will have two different forms of support that I will offer to them. The first is simply a table that will help them to organize the values and allow them to see the relationships between the numbers more clearly. This table can be seen in the attached document titled Lesson 1 [C]. Ideally I would like for them to develop this table on their own but I will have one prepared just in case. This will help the students organize their thoughts and figure out exactly what they are comparing.
If groups are still struggling I will provide them with the second level of support. I will give them a page that shows the possible relationships that they can find between the sides of the triangle and the sin/cos/tan of a given angle. This can be seen in the attached document Lesson 1 [D].
SUMMARIZE/SHARE A:
Once all of the groups have finished “exploring” the trigonometric relationships, the class will reconvene as a whole.
“Okay class! It seems as though everyone has completed the challenge. However, it’s now time to figure out which relationships are the correct relationships! At each group’s station there should be a small whiteboard. I want you to record your findings on this whiteboard and choose a spokesperson from your group who will have to present to the class and convince us that your answers are correct!”
I will have each group present their findings and attempt to convince the rest of the class that the relationships that they found are the correct relationships. Each group will have been given a small whiteboard on which they can record their findings and they can have a spokesperson use this whiteboard to explain to the class what they found out and the process that they used. This will lead to a class discussion in which each group displays their findings and compares them to the rest of the class. As a whole, we will come to a conclusion about the proper relationships and I will explain how this leads to the SOHCAHTOA acronym. Ideally students will be able to establish the three basic ratios and I will present this acronym to them in order to give them an easier way to remember the ratios. I will formalize these rules and present the students with a brief note session. After this activity they should fully understand how the sides of a right triangle are related to the cos/sine/tan of the acute angles of the triangle.
LAUNCH B:
The second LES segment that will occur throughout this lesson includes a “scavenger” hunt on the football field. I will explain to the students that we will be going outside for a scavenger hunt and that they will need their notebooks, a tape measure, and a calculator. Prior to going outside I will explain what is meant by “45 degrees North of East”. Many directions are expressed in this way and I want to ensure that they understand the idea. Next, I will have the students find a partner and each group of partners will receive an envelope.
“Okay class! Now we will be taking a little trip out to the football field! Each group of partners will receive an envelope with a set of directions that you must follow on the field. You will have to apply what you just learned in the previous activity in order to follow the directions”
On the front of the envelope will be a location on the football field from which they must begin. This location will differ for all of the groups. They will not be able to open the envelope until they get to the football field. The students should all be intrigued by this activity and curious to see what’s in the envelope.
EXPLORE B:
Inside each of the student’s envelopes will be a set of 10 directions for them to follow. The students must follow these 10 directions starting from the designated location and once they are done they must record what yard line they are closest to (specifying which side of the field as well) and the distance they are from the closest sideline. They will bring this information back to me and I will check to see if they finished in the correct location (i.e. if they followed the directions properly) I will have calculated where each envelope should end up in order to make this process quicker. If they did not finish in the correct location, then they must repeat the sequence to determine where they made a mistake. The directions will be similar to the following (walk 10 yards due north OR walk 15 yards @ 30 degrees North of East). Since the students do not have a protractor they can’t walk in a diagonal line. Therefore, they must find the components of the given “vectors” in the direction parallel with the yard lines and perpendicular to the yard lines (x and y). To do this they need to use what they learned about sine and cosine in the first portion of the lesson. An example of a set of directions can be seen in the document Lesson 1 [E].
If students are struggling I will ask them some guiding questions:
“If you do not have a protractor to measure out an angle how can we accurately determine where to move?”
“Remember that we can walk parallel and perpendicular to the field without needing to use a protractor”
“With the angle and the distance that is given in each of the directions is there any way to form a right triangle?”
“Which side of the right triangle does the distance that is given represent?”
NOTE: For simplicity, “north” will be specified to be in the direction of the field goal that is closest to the true north direction. I will make this clear to my students.
If groups finish early I will have more challenging envelopes for them to try and they can also exchange envelopes to get more practice.
If groups are really struggling with this assignment I will follow the directions with them to ensure that they are using the proper equations and techniques. I may also pair these groups up with groups that were successful and have them assist one another.
SUMMARIZE/SHARE B:
Each group will return to me on the field once they finish an envelope and I will provide them with a new one. Once I feel as though all of the students completed the activity and found it beneficial we will return to the classroom and talk about the process that was used throughout the investigation. I will ask them what they found challenging and I will clear up any misconceptions that students may have had.
Launch A:
During the Launch phase of the first activity, the student will be informally assessed and I will determine whether or not they are prepared to begin working on the first investigation. I will walk around to the different groups during this first activity and closely observe the work that they are doing. If they seem confused on dealing with right triangles and on identifying the various sides of a right triangle (especially when the triangle is positioned differently) then I may need to spend more time in this launch phase in order to build up the prerequisite knowledge that will be necessary for the lesson.
Explore A:
During the first investigation I will again assess the students informally on their understanding of the content of the lesson. I will observe each of the groups during this investigation and I will ask them the questions that I have detailed above. Based on their responses and on their ability to complete the student investigation, I will be able to determine how well they understand the content of the lesson and how prepared they will be to apply these concepts to other applications. I will keep a journal during the class period and record notes about each student in the class to ensure that they understand the concepts that are being investigated and to ensure that they have the necessary skills that will be required to move forward in the curriculum. I want to ensure that all of the students demonstrate an understanding about how the sine, cosine, and tangent of an acute angle in a right triangle are related to the sides of the triangle. No grade will be assigned to each student but I will use the assessments that I make to determine whether or not more instruction of the concepts will be necessary. If the class as a whole struggled with the material I will plan a follow up lesson to help them understand the material more fully.
Summarize A:
During the summarize phase of the first activity the students will have to explain their thought process as well as their results. This will give me an opportunity to further gauge their understanding of the material. If a student can fully explain what they found and how it relates to right triangles in general then I can conclude that they have a firm grasp on the necessary concepts. I can also assess the students in a formal way by determining if the results that they found are accurate.Launch/Explore/Summarize B:
During this second activity I will informally assess the students on their ability to apply what they learned in the first activity to a real world application. I will observe them as they follow the directions given to them on the football field and I will determine if they truly understand how to use sine, cosine, and tangent to determine information about a given right triangle (break a vector into its components). Again, if I do not feel as though the class (as a whole) is proficient at the task then I will certainly prepare a follow up exercise to provide the students with more practice on the material. I can also assess them formally by determining if they completed the set of directions properly and wound up in the location that they were supposed to finish in.
This section to be completed only if lesson plan is implemented.
Student Work Sample 1 – Approaching Proficiency:
Student Work Sample 2 – Proficient:
Student Work Sample 3 – Exceeds Proficiency:
Lesson Implementation: Was not able to implement this lesson yet.